In Mathematics, the adjective "*linear*" is used in two main senses:

- A function φ on a vector space V over a field
**F** is called *linear* iff:
- ∀x,y∈V: φ(x+y)=φ(x)+φ(y)
- ∀x∈V,a∈
**F**: φ(a*x)=a*φ(x).

For a function over the field **R**, the second condition will follow if φ is also known to be continuous.

I've intentionally said nothing about the *range* of φ. When the range is **F**, φ is a linear functional; when it is another vector space W, φ is a linear transformation. The adjective might be applied in other situations (even when not working on a vector space!), too. For instance, a little-known fact is that perimeter is linear for convex shapes in **R**^{2}!

Confusingly, a function l:**R**->**R**^{k} is *also* called "linear" if it describes a straight line: l(x)=x*a+b.
When b!=0, l is **NOT** "linear" in the first sense! It would more properly be called an affine function. Unfortunately we're stuck with the older name.